How can we design neural networks that allow for stable universal approximation of maps between topologically interesting manifolds? In this talk, we will provide the surprisingly simple answer. By exploiting the topological parallels between locally bilipschitz maps, covering spaces, and local homeomorphisms as well as universal approximation arguments from machine learning, we find that a novel network of the form $p \circ \mathcal{E}$, where $\mathcal{E}$ is a smooth embedding and $p$ a fixed coordinate projection, are universal approximators of local diffeomorphisms between compact smooth submanifolds embedded in $\mathbb{R}^n$. We emphasize the case when the map to be learned changes topology. Further, we find that by constraining the projection $p$, multivalued inversions of our networks can be computed without sacrificing universality. As an application of the problem, we show that the question of learning a group invariant function where the group action is unknown can be naturally reduced to the question of learning local diffeomorphisms when the group action is continuous, finite, and has constant-sized orbits. In this context the novel inversion result permits us to recover orbits of the group action.

We consider the problem of finding the resistors in a network from knowing the power that they dissipate under loads imposed at a few terminal nodes. This data could be obtained e.g. from thermal imaging of the network. We use a method inspired by Bal [1] to give sufficient conditions under which the linearized problem admits a unique solution. Similar results are shown for a discrete analogue to the Schrödinger equation and for the case of impedances or complex valued conductivities.

[1] Bal, Guillaume. Hybrid inverse problems and redundant systems of partial differential equations, Inverse problems and applications 619: 15-48, 2014.

I will present a collaboration with applied mathematicians and civil engineers from the mathematical point of view. We worked on the problem of imaging water supply pipes for problem detection (is there a problem? where is the problem? how severe is the problem?). I will talk about the one-dimensional setting and also present a reconstruction algorithm for tree networks. The problem is modeled mathematically by a quantum tree graph with fluid pressure and flow, and the pipe's internal cross-sectional area as an unknown. The method is based on a simple time reversal boundary control method originally presented by Sondhi and Gopinath for one dimensional problems and later by Oksanen to higher dimensional manifolds.

Simple interactions between particles, people, or neurons give rise to astoundigly complex dynamics on the underlying interaction graphs. I will describe a class of models for dynamical systems on graphs which seems to provide an accurate description for a variety of phenomena from diverse domains. I will then show how this "deep graph dynamics prior" leads to an algorithm to reconstruct the unknown interaction graph when only the dynamics are observed. Potential applications in physics, publich health, Earth science, and neuroscience are important and numerous.